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This paper studies the power allocation problems for a multiple-input multiple-output (MIMO) antenna system with various partial channel state information (CSI) feedback. Through the derivation of new upper bounds of the ergodic capacity for both the cases of channel mean and covariance information feedback, we devise simple closed-form power allocation solutions for the MIMO systems with two transmit antennas. These results are then extended to systems with any number of antennas at both sides. Unlike previous approaches which require computationally complex numerical optimizations over random processes, our solutions in closed-form offer a water-filling interpretation and perform nearly the same as the global optimum.